Cyclic Homology of Strong Smash Product Algebras

Abstract

For any strong smash product algebra $A\#_{_R}B$ of two algebras $A$ and $B$ with a bijective morphism $R$ mapping from $B\ot A$ to $A\ot B$, we construct a cylindrical module $A\natural B$ whose diagonal cyclic module $Δ_{\bullet}(A\natural B)$ is graphically proven to be isomorphic to $C_{\bullet}(A\#_{_R}B)$ the cyclic module of the algebra. A spectral sequence is established to converge to the cyclic homology of $A\#_{_R}B$. Examples are provided to show how our results work. Particularly, the cyclic homology of the Pareigis' Hopf algebra is obtained in the way.

Journal für die Reine und Angewandte Mathematik (2010)